Chapter 1.4, Problem 24E

a.

To determine

find solution values correct to two decimal places.

Answer to Problem 24E

  $\boxed{1.2}$

  $\boxed{-3.29}$

Explanation of Solution

Given information:

The equation $\cos x=x$ has exactly one solution.

Use a graph to show that the equation $\cos x=0.3x$ has three solutions and find their values correct to two decimal places.

Calculation:

Consider the equation $\cos x=0.3x$

The solutions of the equation $\cos x=0.3x$ are the $x$ -coordinates of the point of intersection of the curves $y=\cos x$ and $y=0.3x$ .

From the below, we see that there are three solutions and it lies between $-4$ and 2.

  

In the above graph, there are three point of intersections between two graphs, so we located each point of intersection by the numbering $(1)$ , $(2)$ and $(3)$ .

Now we have to find the values of point of intersections up to two decimal places.

To find the point of intersections of $(1)$ , from the figure(a) we see that it lies between $1$ and $2$ . Zooming in to the viewing rectangle $[1,1.5]$ by $[0.25,0.5]$ . In figure (b), by moving the cursor to the intersection point of the two curves, we see that the solution of the equation is about $\boxed{1.2}$ .

  

To find the point of intersections of $(2)$ , from the graph we see that it lies between $-3$ and $-2$ . Zooming in to the viewing rectangle $[-2.4,-2.2]$ by $[-1,-0.5]$ . In figure , by moving the cursor to the intersection point of the two curves, we see that the solution of the equation is about $\boxed{-2.36}$ .

  

To find the point of intersections of $(3)$ , from the figure(a) we see that it lies between $-4$ and $-3$ . Zooming in to the viewing rectangle $[-3.3,-3.2]$ by $[-1,-0.96]$ . In figure by moving the cursor to the intersection point of the two curves, we see that the solution of the equation is about $\boxed{-3.29}$ .

  

Hence, we conclude that there are two points of intersection because there are two solutions, both within our viewing window.

b.

To determine

Find the approximate value of $m$ .

Answer to Problem 24E

  $\boxed{m\approx 0.34}$

Explanation of Solution

Given information:

the equation $\cos x=x$ has exactly one solution.

Find an approximate value of $m$ such that the equation has $\cos x=mx$ exactly two solutions.

Calculation:

Need to find the approximate value of $m$ such that the equation has $\cos x=mx$ exactly two solutions.

The following graphs shows the different values of $m$ of the equation $y=mx$ ,and the intersecting point of two curves $y=\cos x$ and $y=mx$ that has exactly two solution.

The value of $m$ is obtained by using trial and error method.

  

From the above graph, we observe that the value for $\boxed{m\approx 0.34}$ .

The intersecting points has exactly two solutions as shown in the below graph.

  

Hence,we conclude the value for $\boxed{m\approx 0.34}$